Teaching Mathematics using an approach that is both conceptual and procedural - ​

Nine keys to the hybrid conceptual-procedural approach

Richard Andrew

Contains references to online courses.

There has been a long-running debate about whether a conceptual approach is better for teaching mathematics or whether teaching procedurally is superior. The debate divides mathematics educators into two opposing camps - the proceduralists vs the conceptualists. The arguments from each camp have tended to be mutually exclusive.

Below are nine key points explaining why we can no longer argue that the ideal approach to mathematics education is either procedural or conceptual and that a hybrid approach is required.

Some of the nine keys require expanding. Links to the expansions are provided throughout.

Key One​Both procedural and conceptual approaches to the teaching of mathematics contain necessary prerequisites in order for students to be successful with high school mathematics.

Routines and Procedures

Key TwoThe necessary prerequisite to learning mathematics which is best provided by a quality procedural approach is the explicit teaching of routines and procedures.

Conceptual Understanding

Key ThreeThe necessary prerequisite to learning mathematics which is best provided by a quality conceptual approach is conceptual understanding in students (students actually understanding what it is they are working on in class).Click to read: Key #3 unpacked

We need to reference ‘constructivism’

Let me offer a few ‘feet-on-the-ground’ thoughts on constructivism.

My take is that constructivism is a description of the process by which most of us learn anything of real importance. By ‘real importance’, I mean deciding we need a new skill and so we initiate a self-directed, learning-by-doing path to master that skill. Examples? Learning to sew, to build a shed (or house), mastering the art of gardening, painting and so on.

So constructivism could be seen to be learning via self-initiated, practical exploration. I would say ‘Montessori’ falls under the umbrella of constructivism.

Key FourThe procedural vs conceptual debate is flawed because the common belief (that the two approaches are mutually exclusive) is a misconception. A hybrid approach is possible which includes the essential prerequisites from both approaches - the explicit teaching of procedures as well as ‘enabling students to understand what it is they are working on' (from the time they first start working on it).

Authentic engagement of students

Key FiveThe authentic engagement of students is a fundamental pre-requisite for the learning of mathematics but it rarely features as part of the procedural vs conceptual debate. Unless students are authentically engaged in what they are working on - i.e. immersed in activities, understanding the mathematics and owning their learning - then their ability to learn is severely handicapped. A well-structured conceptual approach is superior when it comes to authentically engaging students i.e. immersing them in activities, having them understand what it is they are working on and owning their learning.​Click to read: Key #5 unpacked

The hybrid approach

Key SixThe hybrid conceptual approach is a highly-structured, mostly student-centred, conceptually-based approach but one that also incorporates the explicit teaching of procedures.

There are two important principles required for the hybrid conceptual approach to be successful:

The hybrid conceptual approach must be (predominantly) student-centred. Yes, it is highly structured. Yes, the teacher still has control over the direction of the learning. But it cannot be driven by the teacher-centredness that drives the procedural approach.

The first aim of any teacher using the hybrid conceptual approach is to have students understand what it is they are doing within an activity. The second aim of the teacher - also important but nevertheless it must follow the first - is the teaching of procedures.

A paradigm shift

Key SevenAs has already been suggested, adopting the hybrid conceptual approach requires a very different pedagogy to a traditional procedural approach. Although the hybrid conceptual approach is easy to administer once sufficient experience has been gained, the transition from a traditional procedural approach to the hybrid conceptual approach requires time and guidance.Key EightThe transition requires a paradigm shift in the teachers’ thinking and classroom management. Ideally, teachers making the shift will receive some quality, ongoing professional guidance which requires them to implement a range of well-structured, student-centred, conceptually-based activities.

Course Information:

Some points in closing

Some final points need mentioning with regard to adopting the hybrid conceptual approach.

A conceptually-based activity is NOT synonymous with ‘lots of equipment’

A common misconception is that a conceptual approach to teaching mathematics requires hands-on activities with lots of equipment. This is not the case. A hybrid-conceptually-based activity is NOT synonymous with ‘lots of equipment’. This is fortunate because the successful running of equipment-heavy lessons is extremely difficult with most high school students. The example given when unpacking Bullet #3 (the rounding of decimals) did likely involve some equipment but equipment that was handled only by the teacher and perhaps a few students in conjunction with quality, targeted, Socratic questioning.

Note that while the decimal rounding example was relatively easy to articulate it would be almost impossible to explain the hybrid conceptual approach for an entire unit. It would require multiple examples each with several sequenced pages containing explanations, experience-based stories and videos as well as sufficient time for teachers to assimilate the information and implement the approach. The name for such an arrangement of information is ‘comprehensive online course’. (!)

A thought-provoking checklist/survey

For students to effectively learn mathematics they need to understand the mathematics they are working on and they also need to learn routines and procedures.

Students gain understanding best via a quality conceptual approach.

Students learn procedures best via a quality procedural approach.

Therefore a hybrid approach is required that combines the best of both. We call this the hybrid conceptual approach.

Teachers using the hybrid conceptual approach have as their primary, initial aim to have students understand the activities they are working through.

The explicit teaching of procedures occurs as an important, secondary focus.

Authentic student engagement is another prerequisite for the effective learning of mathematics.

The hybrid conceptual approach provides an excellent basis for the authentic engagement of students because it is highly-structured, student-centred and conceptually-based.

The student-centred nature of the approach increases the level of student spread. To deal with increased spread teaching segments predominantly occur via mini-lessons.

A reason for student-centred learning and the use of mini-lessons is that it is much more effective to teach students some content when they are ready for it and asking for it than to teach the content to them when we decide we need to teach it to them.

It is important to know that the successful adoption of a hybrid-conceptual approach (or any conceptual approach) requires a very different pedagogy to that used for a traditional procedural approach.

Learn Implement Share has two courses specifically designed to guide teachers into better-engagement of students via the hybrid-conceptually-based approach.

‘Hybrid’ is just for this article!

The term ‘hybrid’ has been woven into this article as a way of high lighting that the strengths of the procedural approach can be incorporated into a quality conceptual approach. However, in Learn Implement Share courses, we don’t refer to a ‘hybrid’ conceptual approach - we simply call it a quality conceptual approach which, in our view, naturally incorporates the explicit teaching of procedures.

Over to you

Your input is welcome. It is a massive challenge to attempt to nail down the principles of the hybrid approach in an article. I’d love to know what questions you have, what surprised you, what was new for you, what you have doubts about and so on.